MMA707- Analytical Finance I

Curran Model to Calculate the Value of Asian Options

Emmanuel N

JAN RÖMAN

Department of Mathematics and Physics

Mälardalen University

SE-721 23 Västerås, Sweden

ABSTRACT

Asian options are options in which the underlying variable is the average price over a particular period of time. Because of this reason, Asian options have a lower volatility and hence making them cheaper relative to their European counterparts. They are commonly traded on currencies and commodity products which have low trading volumes. Industrial consumers are highly exposed to volatility in the cost of electricity. This case study focuses on how a business hedges the electricity price volatility by using the average-price options. Using Monte Carlo Simulation, we calculated the option price at the end.

TABLE OF CONTENT .

INTRODUCTION ................................................................................................................................... 4

1.1 Definition . ............................. 4

1.2 Difficulties . .............................. 4

APPLICATION AND METHODS ........................................................................................................... 5

2.1-Geometric method (Kemna & Vorst) .. ..................... 5

2.2 Arithmetic Rate Approximation (Turnbull & Wakeman) . ............... 7

2.3 Arithmetic Rate Approximation (Levy) ....................... 9

2.5 Binomial Method & Trinomial Trees . ...................... 12

2.6 Finite Differences Method . ....................... 13

2.7 Arithmetic Rate Approximation (Monte Carlo Simulation) . ................... 13

A Case Study in Electricity Markets ..................................................................................................... 14

3.1 Empirical Results . ......................... 16

Table 1: Statistics Output from Crystal Ball Simulation . ............................................................................................................................................................... 17

Table 2 Asian Option Price Calculation ............................................................................................ 17

CONCLUSION ..................................................................................................................................... 18

REFERENCES: ..................................................................................................................................... 19

APPENDIX ........................................................................................................................................... 20

4

INTRODUCTION .

Asian Options were originally used in 1987 when Banker's Trust Tokyo office used them for calculating average options on crude oil contracts, and give the name Asian option. There are no known solutions for arithmetic options. It is not possible to analytically evaluate the sum of the correlated lognormal random variables. Asian Options usually referred to as average options, whose payoff is a function of the average of asset (or underlying) prices over a particular time period, and, thus, are dependent upon the particular path of asset prices. In this article a presentation of some of the applications of Asian Options will be shown, then a brief description of methods to calculate Asian Options and their hedge parameters will be shown, followed by some information of the difficulties encountered when attempting to calculate Asian Options and it will be explained is mainly due to the numerous and varied different existing methods. We will study below how in competitive electricity markets, the price behavior of power is extremely volatile. 1.1 Definition .

Asian options are financial contracts giving the holder the right to buy a certain asset for a pay-off price related to its average price during a particular time period (expiry date). The averaging can be arithmetic or geometric, the average price can be either discretely sampled or continuous sampled. Because of this fact; Asian options have a lower volatility and hence making them cheaper relative to their European counterparts. They are commonly traded on currencies and commodity products which have low trading volumes. There are two types of Asian options available in the market, average rate options and average strike options. The average rate (call) option or ARO payoff at maturity is defined to be the difference (if positive) between the average of (asset) prices recorded over a particular time interval and a particular strike price. However, the average strike option or ASO, as Levy stated, pays the difference (if positive) between the asset price at expiry and the average of asset prices recorded over a particular time interval. Moreover, Geman and Yor state that, the ASO is less used in practice.

1.2 Difficulties .

The difficulties in calculating Asian options are within a Black-Scholes framework, where the price of the underlying asset at any future time is modeled by a lognormal density function, the payoff at expiration, using an arithmetic average, is necessarily a sum of lognormally distributed random variables. Moreover, since the density of the sum of these components has no explicit representation, it’s not possible to produce a closed form solution for calculating arithmetic Asian options. However, since the product of lognormally distributed random variables is itself lognormal, various methods can produce a closed form solutions to this problem. Yet, in as much as the geometric average is less than or equal the arithmetic average on a set of numbers, these techniques tend to underpriced the value of an arithmetic Asian option. Furthermore, using put-call parity to determine the value of an arithmetic average rate put option, the geometric approach tends to overprice the value of an Asian put as well.

5

APPLICATION AND METHODS ..

There are many reasons why Asian options have become popular in the market. The main reasons, as Levy states, first, a company’s exposure to future price movements are sometimes naturally expressed as exposure to the average of prices in the future. Secondly, average options are less sensitive to movements in the underlying asset price when the option’s life is close to maturity. And thirdly, some accounting standards may require the translation of foreign currency assets or liabilities at an average of exchange rates over the accounting period. In addition, Asian options are commonplace in the currency and energy markets, where a firm that is susceptible to asset price fluctuations could use Asian options to speculate on the average of asset prices over a particular time interval. For example, a Brazilian coffee exporter to the United States who is concerned with appreciation of the Brazilian cruzado verses the US dollar might want to hedge the value of the cruzado against future dollar receipts. Moreover, an electric company, whose energy demands increase during the winter months, might want to hedge the costs of coal consumption in January over a particular time interval, say October through December, trying to reduce the impact of dramatic price fluctuations during the time of greatest consumption. In following years, because of the popularity in the market, there has been a lot of interest and effort spent in evaluating average rate options. The methods of valuation of Asian options can be put into the following distinct classes of analytical approximation:

• Geometric method (Kemna & Vorst) • Arithmetic Rate Approximation (Turnbull & Wakeman) • Arithmetic Rate Approximation (Levy) • Binomial Method & Trinomial Trees • Finite Differences Method

• Monte Carlo simulations method with variance reduction techniques. • The partial-differential equation (PDE) approach.

2.1-Geometric method (Kemna & Vorst) ..

In 1990, Kemna & Vorst put forward a closed form of evaluating geometric averaging options by altering the volatility and cost of carry term. Geometric averaging options can be priced through a closed form analytic solution because of the reason that the geometric average of the underlying prices follows a lognormal distribution as well, whereas with arithmetic average rate options, this condition is not possible. The payoff of geometric Asian options is given as:

−− ∏

=−

nn

iiAsian SXCputPayoff

/1

1

,max

−

− ∏

=

− XCPayoff

nn

iiCallAsian S

/1

1

max

6

The payoff of arithmetic Asian Options is given as:

=∑

=− X

r

S

Payoff

n

r

ij

callAsian1,0max

=∑

=− X

r

S

XPayoff

n

r

ij

putAsian1,0max

To expand on arithmetic averaging, this is seen as being the sum of the sampled asset prices divided by the number of samples:

n

SSSAvg

n

A

,.....,21

And for geometric averaging, the average is taken.

nnG SSSAvg ,.......21=

Where the nth root of the sample value multiple together is taken. The payoff functions of Asian Options are given as An average price Asian ( )( )XSnMaxV A −= ,0

And average strike Asian ( )( )AT XSnMaxV −= ,0 Where n is a binary variable which is set to 1 for a call, and -1 for a put .Asia Options can be both European style and American style exercise. The solution to the Geometric Asian Option for call and put is given as: )()( 2

)(1

))((dNXedNSeC

tTrtTrb

G

−−−− −=

And )()( 1

))((2

)(dNSedNXeP

tTrbtTr

G

−−−− −= Where N(x) is the cumulative normal distribution function of:

T

TbX

S

d

A

A

σ

σ )5.0(ln2

1

++

=

7

T

TbX

S

d

A

A

σ

σ )5.0(ln2

2

−+

=

Which can be simplified as? Tdd Aσ−= 12

The adjusted volatility and dividend yield are given as:

3

σσ =A

−−=

62

1 2σDrb

σ is the volatility, r is the risk free rate of interest and D is the dividend yield.

2.2 Arithmetic Rate Approximation (Turnbull & Wakeman) .

. In 1991, Turnbull and Wakeman (TW) suggested an approximation by making use of the fact that the distribution under arithmetic averaging is approximately lognormal, and they put forward the first and second moments of the average in order to evaluate the option. As there are no closed model solutions to arithmetic averages because of inappropriate use of the lognormal assumption under this form of averaging and a number of approximations have been put forward in literature. The analytical approximations for a call and under TW are given as

8

)()( 1)((

222 dNSedNXeP

TrbrT

TW

−− −≈

Where

2

2

2

2

)5.0(ln

T

TbX

S

d

A

A

σ

σ−+

=

212 Tdd Aσ−=

Where 2T is the time remaining up to expiry date. For average options which have already

begun their averaging time, T is t (that is the original time to maturity), if the averaging time has not yet begun, 2T became τ−T .

The adjusted volatility and dividend yield are as:

bT

MA 2

)ln( 2 −=σ

T

Mb

)ln( 1=

To generalize the equations, we assume that the averaging time has not yet begun and given the first and second term moment as:

))((

)()(

1tTDr

eeM

DrDr

−−

−=

Γ−Γ−

+−−

+−−−+

−+−+−=

Γ−Γ+−Γ+−

2

)(

22

))(2(

222

))(2(

2)(2

1

))((

2

))(22)((

222

σσσσ

σσ

Dr

e

DrtTDr

e

tTqrDr

eM

DrDrLr

If the averaging time has already begun, we must adjust the strike price as:

AvgA ST

TTX

T

TX

2

2

2

)( −−=

Where T is the original time to maturity, 2T the remaining time to maturity, X the original

strike price and AvgS is the averaging asset price. Notes that if Dr = the formula will not give

a solution and this was state by Haug in 1998.

9

2.3 Arithmetic Rate Approximation (Levy) . Another analytical approximation was puts forward by Levy which suggested giving more accurate results than the TW approximation. We shall look at the differences below. The approximation to a call is given as: )()( 21

2 dNeXdNSCrT

ZZLevy

−−≈ And through put-call parity, we get the price of a put as:

2rT

ZZLevyLevy eXSCP−+−≈

Where

−= )ln(

2

)ln(11 ZX

L

Kd

Kdd −= 12 And

)()(

22 rTDT

Z eeTDr

SS

−− −−

=

T

TTSXX AvgZ

2−−=

[ ])ln(2)ln( 2 ZSrTLK +−=

2T

ML =

Dr

e

Dr

e

Dr

SM

TDrTDr

−

−−

+−+−=

−−+− 1

)(2

2 222 )(

2

1))(2(

2

2

σσ

σ

Where the variables are the same as defined under the TW approximation. For the various input, we compared the price an Asian call under the TW approximation to that of Levy approximation

10

Asset Price: 100, Average Price: 95 D: 5%, r: 10%, V: 15%, T: 0, T1: 1, T2: 0.5 TW Levy Absolute X Call Value Call Value Error 95 3.202859 3.199390 0.0034690 96 2.444752 2.440545 0.0042066 97 1.787605 1.782873 0.0047318 98 1.246971 1.242086 0.0048849 99 0.827130 0.822518 0.0046122 100 0.520494 0.516509 0.0039841 101 0.310270 0.307114 0.0031558 102 0.175088 0.172788 0.0022995 103 0.093529 0.091982 0.0015470 104 0.047316 0.046352 0.0009645 105 0.022689 0.022130 0.0005593 We can observe that the absolute differences between the 2 approximations are very small, and that the two values can be said to be similar.

In addition, transposing the 2 Call Values as a function of the strike price give the similarity between the two methods.

11

2.4The partial-differential equation (PDE) approach . .

The partial-differential equation (PDE) controlling the Asian option price V(S, A, t) is:

∂

∂σ

∂

∂

∂

∂

∂

∂

V

tS

V

SrS

V

S

S A

t

V

ArV+ + +

−− =

1

202 2

2

2

The discretely sampled arithmetic average value A at the N-th sampling time is:

A

NS tN i

i

i N

==

=

∑1

1

( )

For the discretely sampled Asian option, the equation is the familiar Black-Scholes equation:

02 2

222

=−∂

∂+

∂

∂+

∂

∂rV

S

VrS

S

VS

t

V σ

Addition jump conditions across the sampling dates have to be satisfied:

1. The average value A satisfies the jump condition at the N-th sampling date:

2. The option value V(S, A, t) satisfies the continuous condition:

A AN

A SN N N− = +−

−1

1

1( )

V S A t V S A tN N( , , ) ( , , )−

− +=1

12

2.5 Binomial Method & Trinomial Trees .

.

Asian options can be value using lattice/tree methods. At any point in time on the tree, the value of the option depends on the average of the price that the path has taken. Given the averaging nature of Asian options, a minimum and maximum range at each node must be determined, depending on the path which the underlying asset has taken. The problem is that as the number of nodes on a tree grows, so does the number of averages which must be taken, particularly in the central nodes this is because the number of averages to be taken is exponentially related to the number of possible asset paths. In 1993, Hull & White attempts to solve this problem by adding a state variable to the tree nodes and approximation is undertaken with interpolation techniques in backward induction. The binomial tree can therefore be set up as:

1. The minimum and maximum averages at each time node can be determined as:

i

Sd

Average

i

j

j

iMin

∑−

−

=1

1

)(

i

Su

Average

i

j

j

iMax

∑−

−

=1

1

)(

Where u denotes the size of the up move and d denotes the size of a down move. i is the number of nodes.

2. The approximate average is calculated at each time node. 3. Payoff for each approximate average is determined through means of linear

interpolation 4. Discount backwards towards the first time node:

)(1 ikd

diku

u

rT

ki VPVPeV += −

−

Where k = 1... A. and A is the first time node. uP and dP denote up and down probabilities of

the binomial tree. However, there is the problem that convergence is not guaranteed. In 1997, Chalasani, Jha & Varikooty, came out with a method for the computation of the lower bounds of the Asian option with reasonable accuracy. Their work was base on Rogers & Shi 1995, the authors use a modified choice of random variable z used to estimate the conditional expectation of the option payoff. An improved lower bounds is given, which can be programmed using a binomial lattice.

13

2.6 Finite Differences Method .

In the early mid 90s, several of papers documenting the use of finite differences to solve Asian options where published. Rogers & Shi in 1995, present a method using a one dimensional PDE which can be solved using finite differences. However, their method is prone to problems associated with the diffusion term, especially with lower volatilities and short time to maturities. Andreasen in1998 expand on Rogers & Shi's model by using a change of numeraire to solve the price of Asian options numerically. Example of finite differences methods and the application towards Asian options can also be found in Tavella & Randall in 2000.

2.7 Arithmetic Rate Approximation (Monte Carlo Simulation) .

Several methods using Monte Carlo simulation (MCS) have been developed to price arithmetic Asian options. The analytical approximations by TW, Levy and Curran can all be computed using a simulation method. Monte Carlo simulation can give relatively accurate prices for pricing option, and in the case of Asian options, which are highly path dependent, this method is particularly useful. The control variant technique can be used to find more accurate analytical solutions to a derivative price if there is a similar derivative with a known analytic solution. With this in mind, MCS is then undertaken on the two derivatives in parallel. Given the price of the geometric Asian, we can price the arithmetic Asian by considering the equation:

The estimated value of the arithmetic Asian through simulation is AV and BV is the simulated

value of the geometric Asian, and BV is the exact value of the geometric Asian given above.

14

A Case Study in Electricity Markets . .. . ., . . The price behavior is extremely volatile in electricity markets. In economics and finance, volatility is established in the risks associated with holding assets when there is an uncertain risk associated with the future value of the assets. In competitive electricity markets, pricing are not gradually deregulated, being determined by market participants for each specific interval of the day (e.g. every day ahead), while taking into account various economic and operational factors. In addition, electricity market responds to underlying price drivers that differ dramatically from interest rates and other well developed money markets. Most financial instruments in money markets are traded electronically today. However, electricity responds to the active interchange between producing and using, transferring and storing, buying and selling, and finally consuming actual physical products. In the electricity markets, the supply side concerns not only the storage and transfer of the actual commodity, but also how to get the actual commodity out of the ground. On contrast, the final users consume the commodity with different perspectives. Residential users need power for heating in the winter and cooling in the summer, and industrial users consume energy in order to keep their own production running to avoid the high costs of shutting down and restarting. All market participants have different initiatives with regards to the pricing and determined by different fundamental engine drivers, which in turn significantly affect the behavior of energy markets, causing extremely complex price behavior. These problems lead directly to the need of derivatives contracts. In this study, we are concern on the application of Asian style options in electricity market to discuss how a business hedged or speculate the electricity volatility by using the average price options.

A company is highly exposed to volatility in the cost of electricity. An Asian option is applied based on the average price of a kilowatt hour (or other underlying commodity) over a particular period of time.

A target price of $0.059 was set up based on the past 3-year average price of electricity (Appendix 1). Specifically, the contract terms are as following

� If the average price per kilowatt hour during the next twelve months is greater than this target price, then the counterparty will pay the company the difference.

� If the average price per kilowatt hour during the next twelve months is less than this target price, then the company loses the price it paid for the option.

The question of interests is what the fair price for 1-million kWh per options is.

In this study, we applied Monte Carlo Simulation using Crystal Ball version 7.2 to obtain the proposed option price in one year scenario.

The historical behavior of electricity prices determines the starting point of the modeling process. The model will be based not on the actual prices, but on monthly percent changes in price (or return). To examine the theoretical probability distribution approximating the empirical distribution in these data, the histogram was making as it is shown in Figure 1.

15

Figure 1

Figure 2

It would appear that the percent price changes are approximately normally distributed, and the sample mean and sample standard deviation from these data (0.001768 and 0.073462, respectively) were applied as the mean and standard deviation of the input random variable for the model.

Histogram of Electricity Returns

0

5

10

15

20

25

30

-0.200 -0.175 -0.150 -0.125 -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300

Monthly Price Change

Fre

qu

en

cy

Electricity Price Returns

-0.3

-0.2

-0.2

-0.1

-0.1

0.0

0.1

0.1

0.2

0.2

0.3

0.3

Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99

Months

Mo

nth

ly R

etu

rn

16

3.1 Empirical Results .

The chart frequency indicates that the option is frequently insignificant or meaningless (as evidenced by the tall bar at zero), but that the payout is occasionally $20,000 or more. Remember that the units here are millions of kilowatt hours

Figure 3

To evaluate a fair price, the piece of the simulated sample mean of approximately $3,183 per million kWh (Table 1) is used to calculate the Asian option price as it is shown in Table 2. We are 95% confident that the true fair price is somewhere between $2,843.71 and $3,523.51, which seems like a very wide interval. We could narrow the interval around our estimate by running a longer simulation.

17

Table 1: Statistics Output from Crystal Ball Simulation .

.

Statistic Fit: Beta distribution Forecast values

Trials 1,000

Mean 3,183,61 3,183,61

Median 826,61126 0

Mode 0

Standard Deviation 5,481,86 5,484,60

Variance 30 050 773,59 30 080 854,45

Skewness 2.19 2.19

Kurtosis 8.51 8.49

Coeff. of Variability 1.72 1.72

Minimum – 697,40098 0

Maximum 47,488.53 39,398.75

Mean Std. Error 173,43833

Statistically, a 95% confidence interval is given by:

Table 2 Asian Option Price Calculation

X n

s96.1±

$3,183,61 000,1

60.484,5$96.1±

( )67.171$96.1±

9.339$±

18

CONCLUSION ..

Asian option may be more expensive than the standard option (e.g., options on currencies or oil spreads), and a simple, closed form expression of the Asian option price when the option is in the money. The various methods of analyzing Asian Option have made the instrument popular and easy to use.

19

REFERENCES: ..

http://www.global-derivatives.com/index.php/pricing-models-topmenu- 15:3o,

10/04/10

http://scholar.lib.vt.edu/theses/available/etd-080199-

202859/unrestricted/sudler.pdf 19:00, 10/04/10

http://kevinlanicma.web.officelive.com/Documents/Applying%20Asian%20Option%20to%20Solve%20

an%20Electricity%20Pricing%20Problem.doc

http://74.125.155.132/scholar?q=cache:ZrGiufl_CdEJ:scholar.g

oogle.com/&hl=en&as_sdt=2000

http://www.stat.columbia.edu/~vecer/asiansemi.pdf

http://www.riskglossary.com/link/asian_option.htm

http://www.riskglossary.com/link/asian_option.htm 20:00, 10/04/10

http://janroman.dhis.org/index_eng2.html 20:00 22:00, 10/04/10

20

APPENDIX

Month $/kwh Month $/kwh Month $/kwh

Jan-90 0.0510 May-93 0.0630 Sep-96 0.0580 Feb-90 0.0560 Jun-93 0.0710 Oct-96 0.0570 Mar-90 0.0540 Jul-93 0.0840 Nov-96 0.0550 Apr-90 0.0520 Aug-93 0.0770 Dec-96 0.0550 May-90 0.0520 Sep-93 0.0790 Jan-97 0.0520 Jun-90 0.0570 Oct-93 0.0660 Feb-97 0.0530 Jul-90 0.0670 Nov-93 0.0560 Mar-97 0.0500 Aug-90 0.0640 Dec-93 0.0690 Apr-97 0.0500 Sep-90 0.0640 Jan-94 0.0560 May-97 0.0530 Oct-90 0.0580 Feb-94 0.0540 Jun-97 0.0540 Nov-90 0.0540 Mar-94 0.0530 Jul-97 0.0540 Dec-90 0.0570 Apr-94 0.0560 Aug-97 0.0520 Jan-91 0.0590 May-94 0.0560 Sep-97 0.0500 Feb-91 0.0590 Jun-94 0.0580 Oct-97 0.0550 Mar-91 0.0560 Jul-94 0.0590 Nov-97 0.0520 Apr-91 0.0550 Aug-94 0.0530 Dec-97 0.0480 May-91 0.0570 Sep-94 0.0560 Jan-98 0.0500 Jun-91 0.0620 Oct-94 0.0540 Feb-98 0.0520 Jul-91 0.0710 Nov-94 0.0520 Mar-98 0.0470 Aug-91 0.0690 Dec-94 0.0540 Apr-98 0.0510 Sep-91 0.0690 Jan-95 0.0560 May-98 0.0490 Oct-91 0.0630 Feb-95 0.0580 Jun-98 0.0520 Nov-91 0.0550 Mar-95 0.0560 Jul-98 0.0520 Dec-91 0.0580 Apr-95 0.0580 Aug-98 0.0510 Jan-92 0.0580 May-95 0.0580 Sep-98 0.0510 Feb-92 0.0580 Jun-95 0.0590 Oct-98 0.0470 Mar-92 0.0580 Jul-95 0.0600 Nov-98 0.0470 Apr-92 0.0580 Aug-95 0.0590 Dec-98 0.0450 May-92 0.0600 Sep-95 0.0590 Jan-99 0.0450 Jun-92 0.0690 Oct-95 0.0580 Feb-99 0.0480 Jul-92 0.0800 Nov-95 0.0570 Mar-99 0.0390 Aug-92 0.0750 Dec-95 0.0570 Apr-99 0.0490 Sep-92 0.0740 Jan-96 0.0550 May-99 0.0470 Oct-92 0.0650 Feb-96 0.0550 Jun-99 0.0500 Nov-92 0.0580 Mar-96 0.0550 Jul-99 0.0520 Dec-92 0.0620 Apr-96 0.0550 Aug-99 0.0520 Jan-93 0.0600 May-96 0.0560 Sep-99 0.0510 Feb-93 0.0610 Jun-96 0.0580 Oct-99 0.0480 Mar-93 0.0590 Jul-96 0.0580 Nov-99 0.0460 Apr-93 0.0610 Aug-96 0.0580 Dec-99 0.0460